Method for determining critical operating states in a fuel cell stack

ABSTRACT

The invention relates to a method for determining critical operating states in a fuel cell stack, consisting of single cells connected in series, wherein a low-frequency current or voltage signal is applied to the fuel cell stack, the resulting voltage or current signal is measured and the distortion factor thd is determined. According to the invention, the weighted sum of a term dependent on the membrane resistance RM and a term dependent on the distortion factor thd is used to determine an indicator THDAdryout correlating with the drying out of the fuel cell membranes of the fuel cell stack, the membrane resistance Rm being detected by impedance measurement.

The invention relates to a method for determining critical operatingstates in a fuel cell stack, consisting of individual cells switched inseries, wherein a low-frequency current or voltage signal is applied tothe fuel cell stack, the resulting voltage or current signal is measuredand the distortion factor of the measured signal is determined.

Quality assurance requires checking the functionality and theperformance of all cells in the production of fuel cells. This occursaccording to the state of the art by means of measuring the individualcell voltages for example. Due to the greater technical effort howeverthe measurement of individual cell voltages is neither desirable inproduction nor during operation of fuel cells.

In the laboratory, the so-called impedance spectroscopy is also used fordetecting the operating state or “state of health” of the fuel cellstack. In this process, the complex impedance (i.e. impedance locuscurve) of the fuel cell stack is measured over a specific frequencyrange and mostly compared with reference curves.

Depending on the frequency at which typical changes occur in theimpedance curve, it can now be distinguished whether these changesoriginate from the anode, cathode or the membrane of the individualcells. The method is based on the fact that the electrical equivalentcircuit for the fuel cell stack is a series-parallel circuit of low-passelements of the first order, whose cut-off frequencies are situatedsignificantly far apart and the desired selectivity can therefore beachieved.

It is the following effects in the fuel cell stack (e.g. in a PEM fuelcell stack operated with air and H₂) that substantially requiremonitoring.

-   -   Undersupply of oxidizing agents or fuel at the cathode or anode        (hypostoichiometry). Effect: UI characteristic curve drops        already at lower currents.    -   Membrane: Formation of electrical short-circuits or gas        short-circuits. Effect: U_(o) (voltage at current=0) changes.    -   Electrode ageing. Effect: UI characteristic curve drops more        steeply, higher ohmic resistance by corrosion effects.

The disadvantage in pure impedance spectroscopy is the relatively highmeasuring effort. In addition, impedance spectroscopy is time-consumingbecause it is necessary to measure the impedance in the steady-state ineach of the gradually rising frequencies.

“Total Harmonic Distortion Analysis” (THDA) offers advantages in thisrespect, which represents an online diagnostic tool for determining thestate of fuel cell stacks. Parameters can be extracted with a relativelylow amount of measuring effort, which parameters can be used for thefurther calculations of the state variables of the fuel cell stack.

Such a method, which is based on the analysis of the distortion factor,is described in detail in EP 1 646 101 B1, in which a low-frequencycurrent or voltage signal is applied to the fuel cell stack, theresulting voltage or current signal is measured and conclusions aredrawn on the operating state of individual cells of the fuel cell stackfrom at least one change in the harmonic component (or the distortionfactor) of the signal. This allows an online evaluation of theaforementioned measurements with respect to recognizing and classifyingcritical operating states at the system level.

Distortion factor analysis can occur either in the time range by usingfilters (digital or analog filters) or by means of transformation in thefrequency range (application of all types of wavelet transformations,short-time Fourier transformations or fast-Fourier transformations). Theadvantage of frequency transformation is that the signal-to-noise ratiois improved considerably by this transformation, which again increasesthe sensitivity of the measuring method.

It is the object of the invention, on the basis of the method known fromEP 1 646 101 B1, to propose method variations based on the analysis ofthe distortion factor, with which different critical operating states offuel cell stacks can be distinctly detected, such as:

-   -   Hypostoichiometry at the anode/cathode of the stack    -   Drying out of the membranes of the stack    -   Accumulation of water, formation of droplets on the membranes    -   Deviation of the current minimal cell voltage from the average        cell voltage

A first variant of the invention provides that the weighted sum of aterm dependent on the membrane resistance R_(m) and a term dependent onthe distortion factor thd is used for determining an indicatorTHDA_(dryout) correlating with the drying out of the fuel cell membranesof the fuel cell stack, wherein the membrane resistance R_(m) isdetected by impedance measurement.

A second variant of the invention provides that the weighted sum totalof a term dependent on the internal resistance R_(i), a term dependenton the distortion factor thd and a term dependent on the impedanceR_(lm) of the low-frequency signal is used for determining an indicatorTHDA_(low media) correlating with the stoichiometric undersupply of theanode and/or cathode side of the fuel cell stack.

A third variant of the invention provides that the parameters thd_(dif0)and thd_(dif1), as well as the fluctuations fd(V) in the measuredvoltage curve are used for determining an indicator THDA_(liquid)correlating with impermissible water accumulations and dropletformations on the membranes of the fuel cell stack. thd_(dif0) andthd_(dif1) respectively concern linear combinations of the distortionfactors of current and voltage, wherein thd_(dif0) comprises thecomponent of the first overtones and thd_(dif0). the component of thesecond overtones.

The indicator THDA_(dryout) provides a statement in percentage on thedryness state of the membranes in the stack. THDA_(low media) suppliesthe degree of media undersupply at the cathode or the anode (the mediainclude air, hydrogen or methanol in this case for example).THDA_(liquid) indicates the occurrence of undesirable wateraccumulations.

According to a first advantageous further development of the invention,a simplified electrical equivalent circuit of the fuel cell stack can beused for determining an indicator SoH correlating with the ageing of thefuel cell stack, which equivalent circuit at least considers the ohmicresistances of the cathode side and the anode side R₁, R₂, as well asthe double-layer capacitances C₁, C₂ on the anode and cathode sides andthe inductance L_(m), wherein the parameters for an equation system forthe variables R₁, R₂, C₁, C₂, L_(m) to be determined is set up fromimpedance measurements in at least three measuring frequencies, andwhose solution is used at least partly for calculating the indicatorSoH.

Three measuring frequencies are preferably selected for the calculationin which the impedance curve of the simplified equivalent circuitsubstantially coincides with the real impedance curve of the fuel cellstack.

According to a second advantageous further development of the invention,an artificial neural network (Artificial Neural Network, ANN) can beused for determining an indicator avg-min which correlates with thedeviation of the currently minimal cell voltage from the average cellvoltage, wherein measured quantities derived from the distortion factoranalysis THDA and impedance values derived from the real and imaginarycomponent of the applied voltage signal are used as input quantities ofthe neural network, and wherein the neural network is trained by meansof signals from individual cell voltage measurements for determining theinternal network parameters.

This approach can be expanded in a modular fashion in that the trainingdata record is supplemented by parameters from physical models. Theprecision of the simulation can thus be improved.

A double-layer feed-forward artificial neural network FFANN(Feed-Forward Artificial Neural Network) can preferably be used forsimulating the indicator avg-min correlating with the minimum of thecell voltage of an individual cell of the fuel cell stack, wherein theneural network is adjusted by means of a training function, preferablythe Levenberg-Marquardt training function, to the measured valuesdetected by means of individual cell voltage measurement.

The invention will be explained below in closer detail by reference tothe schematic illustrations shown in the drawings, wherein:

FIG. 1 shows an equivalent circuit of a fuel cell stack;

FIG. 2 shows a schematic sectional view of a PEM fuel cell;

FIG. 3 shows the progression over time of the standardized parameter forTHDA_(dryout) in percent with the representation of the stack voltage involt;

FIG. 4 shows the progression over time of the standardized parameter forTHDA_(low media) in percent with the representation of the stack voltagein volt;

FIG. 5 shows the progression over time of the standardized parameter forTHDA_(liquid) in percent with the representation of the stack voltage involt;

FIG. 6 shows the progression over time of the degree of ageing SoH of astack approximately in the middle of the operational lifespan of thestack;

FIG. 7 shows the progression over time of the degree of ageing SoH of astack in percent after some further days of operation of the stack;

FIG. 8 shows the progression over time of the degree of ageing SoH of astack in percent at the end of the operational lifespan of the stack;

FIG. 9 shows a diagram of a double-layer feed-forward artificial neuralnetwork, and

FIG. 10 shows a diagram of a functional record of a THDA diagnostictool.

The equivalent circuit of the fuel cell stack according to FIG. 1 showsthe ohmic resistances of the cathode and the anode (R₁, R₂) and thedouble-layer capacitances C₁, C₂ on the cathode and anode sides, theinductance of the membrane L_(m) and the ohmic membrane resistanceR_(m). The open-circuit voltage of the stack is designated withreference OCV (open-circuit voltage).

FIG. 2 shows a schematic sectional view through a membrane-electrodeunit MEA of a PEM fuel cell, comprising a membrane 11 disposed betweenan anode 10 and a cathode 12. Anode and cathode 10, 12 respectivelyconsist of a diffusion layer 13 and a catalytic layer 14, 15. The supplyof fuel, e.g. H₂, is marked at the anode with reference numeral 16, andthe discharge with reference numeral 16′. The supply of the oxidizingagent (e.g. air) occurs at reference numeral 17 and the discharge at17′.

THDA_(dryout)

From a highly simplified perspective, a PEM fuel cell is composed of twoelectrodes which are separated by a proton exchange membrane (ProtonExchange Membrane (PEM)), a proton-conductive membrane.

PEMs are polymer electrolyte membranes which are permeable to protons(H⁺), which effectively prevent the transport of gaseous reagents suchas oxygen or hydrogen.

In order to meet these requirements, the membrane must have a specificlevel of humidity among other things. If the membrane dries out, itsconductivity decreases and thus also the performance of the fuel cell.In the worst of cases, cracks and holes are produced and the membrane isirreparably damaged. This strongly limits the functionality of the fuelcell and might even prevent it entirely.

It is necessary to optimize various operational parameters such as gastemperatures and relative humidity in order to achieve the best possibleperformance values and longest operational lifespan of the fuel cell.

Different causes can lead to errors in the operation of the fuel cellstack and result in a decrease in performance. A significant drop in theindividual cell voltages can also be seen with decreasing humidificationof the membrane.

Although this drop in voltage can be detected by means of conventionalindividual cell voltage monitoring, said drop cannot be allocatedspecifically to drying out without further indicators.

The application of the analytic algorithm THDA_(dryout) to the rawmeasured values of the THDA measuring instrument offers the possibilityto monitor parameters which allow drawing conclusions on the state ofthe membrane.

The relevant parameter for THDA_(dryout) is the membrane resistanceR_(m) (see FIG. 1). Depending on the adjustment of specificsystem-dependent parameters (α₀,α₁, reference value ref depending on theage of the stack), a polynomial is obtained depending on the membraneresistance R_(m) for the calculation of THDA_(dryout). It can be seenthat a rising membrane resistance is an indicator for increased dryingout of the membrane. Furthermore, the distortion factor (thd, ratio offundamental component to its harmonic component) is used as anadditional indicator for the non-linear behavior of the system response.

$\begin{matrix}{{{THDA}_{dryout} = {{\alpha_{0}\left( \frac{R_{m} - {ref}}{ref} \right)}^{2} + {\alpha_{1} \cdot {f({thd})}}}},} & {{eqn}.\mspace{14mu} 1}\end{matrix}$

with system-dependent weightings α₀,α₁ with 0<α₀,α₁<1, wherein α₀+α₁=1and polynomial and logarithmic function f. The weightings α₀,α₁ arestrongly system-dependent. The use of smoothing functions (slidingaverage, log, . . . ) depends highly on the structure and the componentsof the system because the signal quality can thus be influenced. Theweighting parameters and smoothing functions are adjusted accordingly bycalibration measurements which follow a special test program in order tosupport the highest possible significance of the data. The first term inequation 1 is usually weighted more strongly.

The membrane resistance R_(m) can be extracted from the highestfrequency of the applied signal because the equivalent circuit of thefuel cell stack is a series-parallel circuit of low-pass elements of thefirst order whose cut-off frequencies are clearly far apart and lie muchlower.

This error identification and the characterization of the errorintensity allows optionally correcting the operating parameters andoptimizing them with respect to the longer term.

The sensitivity of the evaluation algorithms is especially advantageous.A decrease in the relative humidity by 10% at a current density of 0.1A/cm² produces a voltage loss of approximately 8 mV per cell. It ispossible by means of the THDA-dryout calculations to recognize andidentify even minute performance losses.

FIG. 3 shows the progression over time of the voltage of a fuel cellstack (dotted line) at constant current (˜210A). The drop in voltage wascaused by drying out of the membrane. The rise in the dryout intensitysimultaneously with the drop in voltage is shown clearly.

THDA_(low media)

Fuel cells convert chemical reaction energy into electrical energy. Twomedia, i.e. fuel (hydrogen) and the oxidizing agent (usually atmosphericoxygen), must be supplied continuously for this purpose. The optimalsupply of media therefore places an important role in the efficientoperation of a fuel cell.

There can be various causes if an undersupply with media occurs. Thecommon consequence is a drop in the performance of the cell or thestack. Serious or prolonged undersupply with media usually leads toirreparable damage to the cell.

The reasons for the undersupply with media can be wrong gasconcentrations or flows among other things. The formation of waterdroplets in gas conduits can also slow down the supply with media.

On the other hand, a lower flow of air leads to a more inadequateremoval of the water, which is produced as the end product in thereduction on the cathode.

If too little fuel (i.e. hydrogen) is available, the chemical reactionslows down and the performance of the fuel cell decreases. If a massiveundersupply of fuel occurs, local overheating and irreparable damage tothe polymer electrolyte membrane will occur.

The voltage drop caused by the undersupply of media can be monitoredwell by means of conventional measuring methods. The specific advantagein the identification of the cause is offered by the THDA analyticalfunction THDA_(low media).

Inadequate supply with media is detected by means of three parameters.Both the internal resistance R_(i) and also the distortion factor thd ofvoltage signal of the system response to the applied signal as well asan impedance R_(lm) of the low-frequency signal play an important rolein this case.

THDA_(lowmedia)=α₁·ƒ₁(thd)+α₂·ƒ₂(R _(i))+α₃·ƒ₃(R _(lm))  eqn. 2,

with system-dependent weightings α₁,α₂,α₃ with 0<α₁,α₂,α₃<1, whereinα₁+α₂+α₃=1 and evaluation functions f₁, f₂, f₃. The internal resistanceR_(i) is obtained as a sum total of the ohmic resistances R_(m), R₁ andR₂ and is calculated online with the following equation:

${R_{i} = \frac{V_{0} - V}{I \cdot n_{cells}}},$

wherein V concerns the stack voltage, I the stack current, V₀ theopen-circuit voltage and n_(cells) the number of the individual cells inthe stack.

The evaluation functions f₁, f₂, f₃ are used for signal smoothing andare highly system-dependent. Said evaluation functions and the weightingparameters can be determined by calibration measurements by following aspecific protocol in such a way that the signal is as noise-free aspossible and the information content is maximized. The focus in eqn. 2is usually on the second and third term.

For the purpose of differentiating between undersupply of fuel andoxidation agent, the combination of several THDA diagnostic channels canbe helpful. If THDA_(low media) rises in combination with THDA_(liquid),there is a high probability of a stoichiometric undersupply on thecathode side, whereas the combination of THDA_(low media) andTHDA_(dryout) allows drawing a conclusion on the undersupply with mediaon the anode side.

FIG. 4 shows the progression over time of the fuel cell stack atconstant current (˜450A). The voltage drop (shown with the dotted line)was induced by step-by-step reduction in the air supply. This rendersthe removal of water on the cathode side more difficult. Theillustration shows the respective rise in the THDA diagnostic channelsof low media (unbroken line) and liquid water (x).

THDA_(liquid)

The correct water management plays an important role in the operation ofa PEM fuel cell. On the one hand, water is produced as a side product onthe cathode and needs to be discharged from there accordingly. On theother hand, water is introduced into the fuel cell as a result of thehumidification of the gases.

Correct humidification is an important aspect in order to ensure optimalfunctionality of the polymer electrolyte membrane. Drying of themembrane rapidly leads to losses in performance and can permanentlydamage a fuel cell. Conversely, excessive humidity is also not optimal.

If water droplets are formed which cannot be discharged, this will alsolead decreases in performance. Such water droplets can reach the gasdiffusion layer and block gas conduits. This may obstruct gas supply andthe performance of the fuel cell stack decreases.

If the flooding of a fuel cell occurs, this can be monitored by the dropin the cell voltage. The allocation of a voltage drop to the causingerror in the cell is enabled by the analytical function THDA_(liquid).

The accumulation of water droplets produces non-linearities and thusharmonics in the voltage response to the applied signal of the THDAmeasuring instrument.

The occurrence of water accumulations in the gas conduits can bemonitored by an examination of the distortion factor. Rapid minorfluctuations in the voltage curve are also used as an indicator. Thefollowing equation is obtained:

THDA_(liquid)=α₀·ƒ(α₁·abs(thd _(dif0))+α₂·abs(thd _(dif1)))+α₃·ƒd(V)  eqn. 3,

wherein the weightings α₀,α₁,α₂,α₃ are system-dependent parameters with0<α₀,α₁,α₂,α₃<1, f is a polynomial or a logarithmic function forsmoothing and filtering the signals, and thd_(dif0) and thd_(dif1)concern a linear combination of the distortion factors of current andvoltage of the two measured channels (current, voltage). The smoothingfunction f and the weightings α₀,α₁,α₂,α₃ are highly dependent on thesystem configuration and can be determined or adjusted by calibrationmeasurements that follow special test programs in order to optimize theprecision and the interpretability of the result of the calculation.

In order to suppress disturbances in the distortion factor, thedifference (thd_(dif0), thd_(dif1)) between the two is formed once thedisturbances have an effect in the current and voltage distortionfactor. Furthermore, the two lowest frequencies of the applied signalare preferably chosen very close to one another because in the case of adisturbance of the distortion factor the other (disturbance-free)frequency can be used.

The term fd(V), i.e. the finite difference of the voltage values, isoptional because these values too can clearly indicate the occurrence ofwater droplets depending on the type of the fuel cell.

FIG. 5 shows the progression over time of the voltage (dotted line) of afuel cell stack at constant current (˜450A). The formation of waterdroplets was induced by the changed operating parameters and a drop involtage was caused. The THDA diagnostic channel “Liquid Water” shows thecorrelation between voltage drop and droplet formation.

State of Health (SoH)—Degree of Ageing

The result of the state of health measurement reflects the degree ofageing of the stack. In this case, a new stack is assigned 100% SoH and0% SoH to a stack which is at the end of its operational lifespan (e.g.90% loss in performance). The following equations can be made by meansof impedance measurement in the simplified electrical equivalent circuit(see FIG. 1).

$Z_{FC} = {{\frac{R_{1}}{1 + \Omega_{1}^{2}}\left\lbrack {1 - {j\; \Omega_{1}}} \right\rbrack} + {\frac{R_{2}}{1 + \Omega_{2}^{2}}\left\lbrack {1 - {j\; \Omega_{2}}} \right\rbrack} + {R_{m}\left\lbrack {1 + {j\; \Omega_{3}}} \right\rbrack}}$

with Ω₁=ω₁R₁C₁, Ω₂=ω₂R₂C₂, Ω₃=ω₃L_(m),wherein ω₁, ω₂, and ω₃ concern the respective angular frequencies.

If the real and imaginary part of the above complex impedance isregarded separately, the following is obtained:

${{Re}\left\{ Z_{FC} \right\}} = {{\frac{R_{1}}{1 + \Omega_{1}^{2}} + \frac{R_{2}}{1 + \Omega_{2}^{2}} + {R_{m}\mspace{14mu} {and}\mspace{14mu} {Im}\left\{ Z_{FC} \right\}}} = {{- \frac{R_{1}\Omega_{1}}{1 + \Omega_{1}^{2}}} - \frac{R_{2}\Omega_{2}}{1 + \Omega_{2}^{2}} + {R_{m}\Omega_{3}}}}$

A simplification in these equations leads to the following:

${{Re}\left\{ Z_{FC} \right\}} = {{\frac{R_{1}}{1 + \Omega_{1}^{2}} + \frac{R_{2}}{1 + \Omega_{2}^{2}} + {R_{m}\mspace{14mu} {or}\mspace{14mu} {Im}\left\{ Z_{FC} \right\}}} \approx {{- \frac{R_{1}\Omega_{1}}{1 + \Omega_{1}^{2}}} - \frac{R_{2}\Omega_{2}}{1 + \Omega_{2}^{2}}}}$

for frequencies between 5 and 10 Hz.

${{Re}\left\{ Z_{FC} \right\}} \approx {\frac{R_{2}}{1 + \Omega_{2}^{2}} + {R_{m}\mspace{14mu} {or}\mspace{14mu} {Im}\left\{ Z_{FC} \right\}}} \approx {{- \frac{R_{1}}{\Omega_{1}}} - \frac{R_{2}\Omega_{2}}{1 + \Omega_{2}^{2}} + {R_{m}\Omega_{3}}}$

for frequencies between 10 and 100 Hz, and

${{Re}\left\{ Z_{FC} \right\}} \approx {R_{m}\mspace{14mu} {or}\mspace{14mu} {Im}\left\{ Z_{FC} \right\}} \approx {{- \frac{R_{1}}{\Omega_{1}}} - \frac{R_{2}}{\Omega_{2}} + {R_{m}\Omega_{3}}}$

for frequencies over 400 Hz.

These equations are prepared for three frequencies and the respectivereal and imaginary parts of the impedances and resolved according to C₁,C₂, L_(m), R₁ and R₂. It was surprisingly noticed that ageing correlatesdistinctly and approximately linearly with the two double-layercapacitances C₁ (on the cathode side) and C₂ (on the anode side) andwith the ohmic resistances of cathode (R₁) and anode (R₂).

Numeric solving of the equation system produces current values for theindividual components. The ohmic resistances are not used for ageingmeasurement but only the double-layer capacitances for the followingpractical reasons:

-   -   Too many ancillary influences, i.e. ohmic resistances, also        correlate strongly with operating parameters such as pressure,        temperature and media stoichiometry.    -   Very low frequencies (e.g. beneath 1 Hz) are unnecessary for the        reliable determination of the ohmic resistances. Disadvantage:        too long measuring times and too significant disturbing        influences which are derived from the load dynamics.

Consequently, only C₁ and C₂ are used for calculating the value. Thefollowing equation is used for this purpose:

$\begin{matrix}{{{SoH} = {{\alpha_{1} \cdot \left( {{100 \cdot \frac{C_{1}}{0.75\overset{\_}{C_{1}}}} - \frac{100}{3}} \right)} + {\alpha_{2} \cdot \left( {{100 \cdot \frac{C_{2}}{0.75\overset{\_}{C_{2}}}} - \frac{100}{3}} \right)}}},} & {{Eqn}.\mspace{14mu} 4}\end{matrix}$

with system-dependent weightings α₁,α₂ with 0<α₁,α₂<1, wherein α₁+α₂=1and the starting values C₁ and C₂ The weighting α₁,α₂ issystem-dependent and it can be determined by initial calibrationmeasurements which of the two terms shall have a higher influence on thetotal result. The second term is usually more meaningful and thusweighted more strongly.

An advantageous method is applied for minimizing errors in ageingcalculation by means of a simplified equivalent circuit instead of knowncomplex and precisely adjusted equivalent circuits. The three measuringfrequencies are selected in such a way that the impedance curve of thesimplified equivalent circuit coincides with the real impedance curve ofthe fuel cell precisely in these three frequencies.

In order to ensure the reliability of the above equations, the usedquantities are smoothed as required by using sliding weighted averaging.This corresponds to a filter that dampens high frequencies.

FIG. 6 shows the progression over time of the state of health of a fuelcell stack which is calculated online. At this point in time, the stackhas reached approximately half of its operational lifespan (relativeage) (˜54% SoH). After a number of days of operation under varioushighly critical conditions, a drop of the SoH to 46% SoH can be observed(see FIG. 7).

FIG. 8 shows a further measurement a few months later, wherein the endof the operational lifespan of the stack was nearly reached (˜1% SoH).

Minimum of the Cell Voltage Avg-Min

The invention further comprises a method for the numerical determinationof the deviation of a cell voltage minimum from the average cell voltagein a fuel cell stack. This method is used as a replacement for anindividual cell voltage measuring device.

It is common practice to carry out individual cell voltage measurementsfor monitoring a fuel cell stack. An individual cell voltage monitoringdevice (Cell Voltage Monitoring CVM) allows monitoring the voltages ofevery single fuel cell of a stack. It is especially important torecognize whether the voltage of individual cells drops. In the case ofa large number of cells, this can no longer be obtained from the totalvoltage of the stack (the sum total of all individual cell voltages).Specific cases of disturbance (critical states) often have an influenceon a low number of cells at the beginning before a significant drop inthe total voltage occurs. In order to recognize this, the deviation inthe cell voltage minima avg-min from the average cell voltage must bemonitored or determined. This quantity thus supplies the deviation ofthe lowest single cell voltage from the average cell voltage. Singlecell voltage monitoring is complex and is an expensive andspace-consuming method and thus only useful within limits for seriesproduction.

The method in accordance with the invention allows monitoring theavg-min signal without requiring access to the individual cell voltagemeasuring data. It concerns a method that is based on the THDA method.Conclusions can be drawn on the operating state of the fuel cell stackby means of measurement and the respective analyses of the systemresponse (voltage or current) by applying a modulated current or voltagesignal. As described further above, the analysis supplies threecharacteristics THDA_(dryout), THDA_(low media) and THDA_(liquid) forrecognizing the state, and also the phase and amplitude of theimpedances at different frequencies. Furthermore, the distortion factorthd of the system response is calculated. The method described belowuses all measurement quantities from THDA (real and imaginary parts ofthe impedances, distortion factors, SNR (Signal to Noise Ratio)) inorder to thus calculate the avg-min signal.

More precise approximations to the avg-min signal can be reached insteps with three expansion stages in which the THDA measurement data areexpanded by new records from physical models. The most importantadvantage is that the precision of the simulation increases with eachstage. Since the amount of calculation will increase in this way, anapproach is made to expand the method in a modular fashion.

The first expansion stage increases the input data record by extremevalues of the gas concentrations. The maximum water and minimum gasconcentrations (O₂, H₂) can be calculated from the mass flows and gasinput concentrations and can be used as further input quantities.

Stage two comprises an expansion in the data records by results of anelectrochemical model. This model calculates pressures, temperatures atthe gas outputs and voltage differences (Vstack-Vmodel) from referencestack data (voltage-current characteristics at different temperaturesand pressures). This information increases the precision in thesimulation of the avg-min signal.

If there are precise flow channel geometries of the fuel cell stack, thedata record can also be expanded by the calculation results of asimplified thermal model. Local membrane temperatures and temperaturesat the anode, cathode and cooling temperatures can be modeled and supplyfurther information as an input for the avg-min algorithm.

An artificial neural network (Artificial Neural Network, ANN) forms thebasis for the final data evaluation and algorithm development. Itconcerns a network of artificial neurons, an abstraction of the naturalneurons occurring in the brain which is used in the area of informationprocessing and artificial intelligence. The areas of use for ANN arenumerous and range from function fitting and classification problems upto recognition of patterns or also time-series analysis. Once thearchitecture of the network has been determined, it is trained by meansof respective training algorithms, i.e. weightings and parameters in theinterior of the network are adjusted.

A double-layer feed-forward ANN (FFANN) with a fixed number (e.g. 10neurons) was used in the hidden layer (see FIG. 9) for the simulation ofthe avg-min signals. All available measurement quantities of the THDAmethod supply the input quantities for the neural network, wherein thenetwork is trained to the fitting of the original avg-min signals(measured by means of the CVM measuring device) by means of theLevenberg-Marquardt training function.

The result (out) of an FFANN constructed for this method can bedescribed by means of the following equation:

out(in)=ƒ₂·(W ₂·ƒ₁(W ₁·in+b ₁)+b ₂)  eqn. 5

“in” concerns the data input vector and f₁, f₂ the transmissionfunctions. The weighting matrixes W₁, W₂ and the bias vectors b₁, b₂ arerespectively optimized during training.

Since the result of a network trained with the entire data record is notalways reliable, the training data quantity S can be divided in asupporting fashion with respect to a threshold value T. The divisionoccurs into two data quantities on the basis of the avg-min signalmeasured for test purposes on the stack. The first quantity S₁ containsall input quantities which correspond to an avg-min signal<T, and thefollowing applies to the second quantity: S₂=S\S₁. This division of thedata into two groups can also be interpreted as a classification into“non-critical” and “critical” situations since many systems will switchoff or be in a critical state when a value drops beneath a minimum valueof the single cell voltage. The selection of the threshold value issystem-dependent. The reason for the division is that the respectiveavg-min values can be simulated better in some cases when one FFANN eachis trained per data quantity.

The data separation thus leads to two FFANNs N₁ and N₂, which aretrained with the respective data quantities and can be described by twomathematical functions out₁ and out₂. The functional set of the THDAdiagnostic tool is expanded with these two new functions (see FIG. 10).

A separation of the data with respect to a system-dependent thresholdvalue occurs online by means of current classification algorithms(Support Vektor Machine, PCA, Nearest Neighbor, Cluster analysis). Anobservation consisting of one respective instance of each measured valueis classified accordingly and then supplies the input quantity for therespective neural network. The degree of the critical state of the fuelcell stack can be estimated depending on the results supplied by thesenetworks and respective further steps such as open-loop and closed-loopcontrol measures can be initiated accordingly.

1. A method for determining critical operating states in a fuel cellstack, consisting of individual cells connected in series, wherein alow-frequency current or voltage signal is applied to the fuel cellstack, the resulting voltage or current signal is measured and thedistortion factor thd is determined, wherein the weighted sum of a termdependent on the membrane resistance R_(m) and a term dependent on thedistortion factor thd is used to determine an indicator THDA_(dryout)correlating with the drying out of the fuel cell membranes of the fuelcell stack, wherein the membrane resistance R_(m) is detected byimpedance measurement.
 2. The method according to claim 1, wherein theindicator THDA_(dryout) correlating with the drying out of the fuel cellmembranes is determined according to${{THDA}_{dryout} = {{\alpha_{0}\left( \frac{R_{m} - {ref}}{ref} \right)}^{2} + {\alpha_{1} \cdot {f({thd})}}}},$wherein α₀+α₁=1 applies, and ref is a reference value for the membraneresistance and f is a polynomial or a logarithmic function.
 3. Themethod according to claim 1, wherein a simplified electrical equivalentcircuit of the fuel cell stack can be used for additionally determiningan indicator SoH correlating with the ageing of the fuel cell stack,which equivalent circuit at least considers the ohmic resistances of thecathode side and the anode side R₁, R₂, as well as the double-layercapacitances C₁, C₂ on the anode and cathode sides, wherein an equationsystem for the variables R₁, R₂, C₁, C₂ is set up, which is determinedby impedance measurements in at least three measuring frequencies and isused for calculating the indicator SoH.
 4. The method according to claim3, wherein the indicator SoH correlating with the ageing of the fuelcell stack is determined from the parameters C₁, C₂ of the double-layercapacitances according to${{SoH} = {{\alpha_{1} \cdot \left( {{100 \cdot \frac{C_{1}}{0.75\; \overset{\_}{C_{1}}}} - \frac{100}{3}} \right)} + {\alpha_{2} \cdot \left( {{100 \cdot \frac{C_{2}}{0.75\; \overset{\_}{C_{2}}}} - \frac{100}{3}} \right)}}},$wherein the parameters disregard the ohmic resistances of the cathodeside and anode side R₁, R₂, α₁+α₂=1 applies and C₁ and C₂ concernstarting values of a new fuel cell stack.
 5. The method according toclaim 3, wherein three measuring frequencies are selected for thecalculation, in which the impedance curve of the simplified equivalentcircuit coincides substantially with the real impedance curve of thefuel cell stack.
 6. The method according to claim 4, wherein threemeasuring frequencies are selected for the calculation, in which theimpedance curve of the simplified equivalent circuit coincidessubstantially with the real impedance curve of the fuel cell stack. 7.The method according to claim 1, wherein a stoichiometric undersupply ofthe anode side of the fuel cell stack is determined by the combinedoccurrence of a rising membrane resistance R_(m) according to theindicator THDA_(dryout) and a deviation of the internal resistance R_(i)from the reference value according to an indicator THDA_(low media),wherein the weighted sum total of a term dependent on the internalresistance R_(i), a term dependent on the distortion factor thd and aterm dependent on the impedance R_(lm) of the low-frequency signal isused for determining the indicator THDA_(low media) correlating with thestoichiometric undersupply of the anode and/or cathode side of the fuelcell stack.
 8. The method according to claim 1, wherein an artificialneural network (Artificial Neural Network) ANN is used for theadditional determination of an indicator avg-min correlating with thedeviation of the minimum cell voltage from the average cell voltage ofthe fuel cell stack, measured quantities derived from the distortionfactor analysis THDA and impedance values derived from the real andimaginary part of the applied current and voltage signal are used asinput quantities of the network, wherein the neural network is trainedby means of signals from the individual cell voltage measurements fordetermining the internal network parameters.
 9. The method according toclaim 8, wherein a double-layer feed-forward Artificial Neural NetworkFFANN is used for simulating the indicator avg-min correlating with theminimum of the cell voltage of an individual cell of the fuel cellstack, and the neural network is adjusted to the measured valuesdetected by means of individual cell voltage measurement by means of atraining function.
 10. The method according to claim 8, wherein thequantity of training data is expanded in a modular manner by calculationresults from physical models.
 11. The method according to claim 9,wherein the training function is the Levenberg-Marquardt trainingfunction.